Is Energy Conserved in General Relativity?

In special cases, yes. In general — it depends on what you mean by "energy", and
what you mean by "conserved".

In flat spacetime (the backdrop for special relativity) you can phrase energy
conservation in two ways: as a differential equation, or as an equation involving
integrals (gory details below). The two formulations are mathematically
equivalent. But when you try to generalize this to curved spacetimes (the arena for
general relativity) this equivalence breaks down. The differential form extends with
nary a hiccup; not so the integral form.

The differential form says, loosely speaking, that no energy is created in any
infinitesimal piece of spacetime. The integral form says the same for a finite-sized
piece. (This may remind you of the "divergence" and "flux" forms of Gauss's law in
electrostatics, or the equation of continuity in fluid dynamics. Hold on to that
thought!)

An infinitesimal piece of spacetime "looks flat", while the effects of curvature become
evident in a finite piece. (The same holds for curved surfaces in space, of
course). GR relates curvature to gravity. Now, even in newtonian physics, you
must include gravitational potential energy to get energy conservation. And GR
introduces the new phenomenon of gravitational waves; perhaps these carry energy as
well? Perhaps we need to include gravitational energy in some fashion, to arrive at
a law of energy conservation for finite pieces of spacetime?

Casting about for a mathematical expression of these ideas, physicists came up with
something called an energy pseudo-tensor. (In fact, several of 'em!) Now, GR takes
pride in treating all coordinate systems equally. Mathematicians invented tensors
precisely to meet this sort of demand — if a tensor equation holds in one coordinate
system, it holds in all. Pseudo-tensors are not tensors (surprise!), and this alone
raises eyebrows in some circles. In GR, one must always guard against mistaking
artifacts of a particular coordinate system for real physical effects. (See the FAQ
entry on black holes for some examples.)

These pseudo-tensors have some rather strange properties. If you choose the
"wrong" coordinates, they are non-zero even in flat empty spacetime. By another
choice of coordinates, they can be made zero at any chosen point, even in a spacetime full
of gravitational radiation. For these reasons, most physicists who work in general
relativity do not believe the pseudo-tensors give a good local definition of
energy density, although their integrals are sometimes useful as a measure of total
energy.

One other complaint about the pseudo-tensors deserves mention. Einstein argued
that all energy has mass, and all mass acts gravitationally. Does "gravitational
energy" itself act as a source of gravity? Now, the Einstein field equations are

Gmu,nu = 8pi Tmu,nu

Here Gmu,nu is the Einstein curvature tensor, which encodes information about
the curvature of spacetime, and Tmu,nu is the so-called stress-energy tensor,
which we will meet again below. Tmu,nu represents the energy due to
matter and electromagnetic fields, but includes NO contribution from "gravitational
energy". So one can argue that "gravitational energy" does NOT act as a source of
gravity. On the other hand, the Einstein field equations are non-linear; this
implies that gravitational waves interact with each other (unlike light waves in Maxwell's
(linear) theory). So one can argue that "gravitational energy" IS a source of
gravity.

In certain special cases, energy conservation works out with fewer caveats. The
two main examples are static spacetimes and asymptotically flat spacetimes.

Let's look at four examples before plunging deeper into the mathematics. Three
examples involve redshift; the other, gravitational radiation.

Very fast objects emitting light

According to special relativity, you will see light coming from a receding
object as redshifted. So if you, and someone moving with the source, both measure
the light's energy, you'll get different answers. Note that this has nothing to do
with energy conservation per se. Even in newtonian physics, kinetic energy (mv^2/2)
depends on the choice of reference frame. However, relativity serves up a new
twist. In newtonian physics, energy conservation and momentum conservation are two
separate laws. Special relativity welds them into one law, the conservation of the
energy-momentum 4-vector. To learn the whole scoop on 4-vectors, read a
text on SR, for example Taylor and Wheeler (see refs.) For our purposes, it's enough to
remark that 4-vectors are vectors in spacetime, which most people privately picture just
like ordinary vectors (unless they have very active imaginations).

Very massive objects emitting light

Light from the Sun appears redshifted to an Earth bound astronomer. In
quasi-newtonian terms, we might say that light loses kinetic energy as it climbs out of
the gravitational well of the Sun, but gains potential energy. General relativity
looks at it differently. In GR, gravity is described not by a "potential" but by the
"metric" of spacetime. But "no problem", as the saying goes. The Schwarzschild
metric describes spacetime around a massive object, if the object is spherically
symmetrical, uncharged, and "alone in the universe". The Schwarzschild metric is
both static and asymptotically flat, and energy conservation holds without major
pitfalls. For more details, consult MTW, chapter 25.

Gravitational waves

A binary pulsar emits gravitational waves, according to GR, and one expects (innocent
word!) that these waves will carry away energy. So its orbital period should
change. Einstein derived a formula for the rate of change (known as the quadrapole
formula), and in the centenary of Einstein's birth, Russell Hulse and Joseph Taylor
reported that the binary pulsar PSR1913+16 bore out Einstein's predictions within a few
percent. Hulse and Taylor were awarded the Nobel prize in 1993.

Despite this success, Einstein's formula remained controversial for many years, partly
because of the subtleties surrounding energy conservation in GR. The need to
understand this situation better has kept GR theoreticians busy over the last few
years. Einstein's formula now seems well-established, both theoretically and
observationally.

Expansion of the universe leading to cosmological redshift

The Cosmic Background Radiation (CBR) has red-shifted over billions of years.
Each photon gets redder and redder. What happens to this energy? Cosmologists
model the expanding universe with Friedmann-Robertson-Walker (FRW) spacetimes. (The
familiar "expanding balloon speckled with galaxies" belongs to this class of models.) The
FRW spacetimes are neither static nor asymptotically flat. Those who harbor no
qualms about pseudo-tensors will say that radiant energy becomes gravitational
energy. Others will say that the energy is simply lost.

It's time to look at mathematical fine points. There are many to choose
from! The definition of asymptotically flat, for example, calls for some care (see
Stewart); one worries about "boundary conditions at infinity". (In fact, both
spatial infinity and "null infinity" clamor for attention — leading to different kinds of
total energy.) The static case has close connections with Noether's theorem (see
Goldstein or Arnold). If the catch-phrase "time translation symmetry implies
conservation of energy" rings a bell (perhaps from quantum mechanics), then you're on the
right track. (Check out "Killing vector" in the index of MTW, Wald, or Sachs and
Wu.)

But two issues call for more discussion. Why does the equivalence between the two
forms of energy conservation break down? How do the pseudo-tensors slide around this
difficulty?

We've seen already that we should be talking about the energy-momentum 4-vector, not
just its time-like component (the energy). Let's consider first the case of flat
Minkowski spacetime. Recall that the notion of "inertial frame" corresponds to a
special kind of coordinate system (minkowskian coordinates).

Pick an inertial reference frame. Pick a volume V in this frame, and pick two
times t=t0 and t=t1. One formulation of energy-momentum
conservation says that the energy-momentum inside V changes only because of
energy-momentum flowing across the boundary surface (call it S). It is "conceptually
difficult, mathematically easy" to define a quantity T so that the captions on the
Equation 1 (below) are correct. (The quoted phrase comes from Sachs and Wu.)

T is called the stress-energy tensor. You don't need to know what that means! —just
that you can integrate T, as shown, to get 4-vectors. Equation 1 may remind you of
Gauss's theorem, which deals with flux across a boundary. If you look at Equation 1
in the right 4-dimensional frame of mind, you'll discover it really says that the flux
across the boundary of a certain 4-dimensional hypervolume is zero. (The hypervolume
is swept out by V during the interval t=t0 to t=t1.) MTW,
chapter 7, explains this with pictures galore. (See also Wheeler.)

A 4-dimensional analogue to Gauss's theorem shows that Equation 1 is equivalent to:

We write "coord_div" for the divergence, for we will meet another divergence in a
moment. Proof? Quite similar to Gauss's theorem: if the divergence is zero
throughout the hypervolume, then the flux across the boundary must also be zero. On
the other hand, the flux out of an infinitesimally small hypervolume turns out to be the
divergence times the measure of the hypervolume.

Pass now to the general case of any spacetime satisfying Einstein's field
equation. It is easy to generalize the differential form of energy-momentum
conservation, Equation 2:

(Side comment: Equation 3 is the correct generalization of Equation 1 for SR when
non-minkowskian coordinates are used.)

GR relies heavily on the covariant derivative, because the covariant derivative of a
tensor is a tensor, and as we've seen, GR loves tensors. Equation 3 follows from
Einstein's field equation (because something called Bianchi's identity says that
covariant_div(G)=0). But Equation 3 is no longer equivalent to Equation 1!

Why not? Well, the familiar form of Gauss's theorem (from electrostatics) holds
for any spacetime, because essentially you are summing fluxes over a partition of the
volume into infinitesimally small pieces. The sum over the faces of one
infinitesimal piece is a divergence. But the total contribution from an interior
face is zero, since what flows out of one piece flows into its neighbor. So the
integral of the divergence over the volume equals the flux through the boundary.
"QED".

But for the equivalence of Equations 1 and 3, we would need an extension of Gauss's
theorem. Now the flux through a face is not a scalar, but a vector (the flux of
energy-momentum through the face). The argument just sketched involves adding these
vectors, which are defined at different points in spacetime. Such "remote vector
comparison" runs into trouble precisely for curved spacetimes.

The mathematician Levi-Civita invented the standard solution to this problem, and
dubbed it "parallel transport". It's easy to picture parallel transport: just move
the vector along a path, keeping its direction "as constant as possible".
(Naturally, some non-trivial mathematics lurks behind the phrase in quotation marks.
But even pop-science expositions of GR do a good job explaining parallel transport.) The
parallel transport of a vector depends on the transportation path; for the canonical
example, imagine parallel transporting a vector on a sphere. But parallel
transportation over an "infinitesimal distance" suffers no such ambiguity. (It's not
hard to see the connection with curvature.)

To compute a divergence, we need to compare quantities (here vectors) on opposite
faces. Using parallel transport for this leads to the covariant divergence.
This is well-defined, because we're dealing with an infinitesimal hypervolume. But
to add up fluxes all over a finite-sized hypervolume (as in the contemplated extension of
Gauss's theorem) runs smack into the dependence on transportation path. So the flux
integral is not well-defined, and we have no analogue for Gauss's theorem.

One way to get round this is to pick one coordinate system, and transport vectors so
their components stay constant. Partial derivatives replace covariant
derivatives, and Gauss's theorem is restored. The energy pseudo-tensors take this
approach (at least some of them do). If you can mangle Equation 3 (covariant_div(T)
= 0) into the form:

coord_div(Theta) = 0

then you can get an "energy conservation law" in integral form. Einstein was the
first to do this; Dirac, Landau and Lifshitz, and Weinberg all came up with variations on
this theme. We've said enough already on the pros and cons of this approach.

We will not delve into definitions of energy in general relativity such as the
hamiltonian (amusingly, the energy of a closed universe always works out to zero according
to this definition), various kinds of energy one hopes to obtain by "deparametrizing"
Einstein's equations, or "quasilocal energy". There's quite a bit to say about this
sort of thing! Indeed, the issue of energy in general relativity has a lot to do
with the notorious "problem of time" in quantum gravity. . . but that's another can of
worms.

References (vaguely in order of difficulty):

Clifford Will, The renaissance of general relativity, in The New
Physics (ed. Paul Davies) gives a semi-technical discussion of the controversy
over gravitational radiation.

Wheeler, A Journey into Gravity and Spacetime. Wheeler's try at a
"pop-science" treatment of GR. Chapters 6 and 7 are a tour-de-force: Wheeler
tries for a non-technical explanation of Cartan's formulation of Einstein's field
equation. It might be easier just to read MTW!)

John Stewart, Advanced General Relativity. Chapter 3
(Asymptopia) shows just how careful one has to be in asymptotically flat
spacetimes to recover energy conservation. Stewart also discusses the
Bondi-Sachs mass, another contender for "energy".

Damour, in 300 Years of Gravitation (ed. Hawking and Israel). Damour heads the
"Paris group", which has been active in the theory of gravitational radiation.